EF Exercises

Exercises 9.9 Exercises

1.

Prove each of the set operation rules in Proposition 9.4.11. Use the provided proof of the first of DeMorgan’s Laws as a model for your proofs.

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Expressing relationships using the symbols of set theory.

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In each of Exercises 2–4, you are given a collection of sets (and possibly some elements of those sets), a collection of symbols, and a collection of statements about those sets and their elements. Use the given symbols to express the given statements in symbolic language.

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Note that there may be more than one correct answer for each statement.

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2.

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Sets:

\begin{aligned} A & = the set of all Augustana students, \\ R & = the set of Augustana students who attend class regularly, \\ S & = the set of Augustana students who study diligently, \\ P & = the set of Augustana students who will pass all their courses. \end{aligned}

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Symbols:

A, R, S, P, R^{c}, S^{c}, P^{c}, \cap, \cup, =, \neq, \subseteq, ⫋, \emptyset .

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Statements:

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All Augustana students who attend class regularly and study diligently will pass all their courses.
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Some Augustana students attend class regularly but do not study diligently.
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Some Augustana students who study diligently will still fail a course.

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3.

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Recall that a square number is an integer which is equal to the square of some integer. (See the introduction preceding Exercise 6.12.17 in Exercises 6.12.)

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Sets:

\begin{aligned} P & = the set of prime numbers, \\ E & = the set of even numbers, \\ S & = the set of square numbers. \end{aligned}

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Symbols:

2, N, P, E, S, N^{c}, P^{c}, E^{c}, S^{c}, \in, \cap, \cup, =, \neq, \subseteq, ⫋, \emptyset, {} .

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Statements:

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$2$ is the only even, prime number.
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There exist odd square numbers.
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No prime number is square.
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No square number is prime.
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It is not true that every natural number is either even or prime.

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4.

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Sets:

\begin{aligned} F & = the set of all functions in a single real variable, \\ C & = the set of continuous functions, \\ D & = the set of differentiable functions, \\ P & = nonnegative functions \\ = {f (x) | f (x) \geq 0 for all x in the domain of f} . \end{aligned}

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Elements:

\begin{aligned} f_{1} (x) & = x^{2} & f_{2} (x) & = | x | & f_{3} (x) & = \tan x \end{aligned}

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Symbols:

F, C, D, P, F^{c}, C^{c}, D^{c}, P^{c}, f_{1} (x), f_{2} (x), f_{3} (x), \in, \cap, \cup, =, \neq, \subseteq, ⫋, \emptyset .

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Statements:

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The function $f_{1} (x)$ is differentiable and nonnegative.
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The function $f_{2} (x)$ is continuous and nonnegative, but not differentiable.
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The function $f_{3} (x)$ is neither continuous nor nonnegative.
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Every differentiable function is continuous.
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Some continuous functions are not differentiable.
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Not every function is continuous.

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Testing set equalitye.

For each of Exercises 5–8, either formally prove the given equivalence of sets (using the Test for Set Equality) or demonstrate that it is false by providing a specific counterexample.

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5.

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A = (A ∖ B) ⊔ (A \cap B)

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6.

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A ∖ (A ∖ B) = B

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7.

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(A \times B) \cup (C \times D) = (A \cup C) \times (B \cup D)

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8.

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A \times (B ∖ C) = (A \times B) ∖ (A \times C)

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9.

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Suppose

Σ

is an alphabet. Prove that

Σ^{*}

is the disjoint union of the subsets

Σ_{0}^{*}, Σ_{1}^{*}, Σ_{2}^{*}, \dots, Σ_{n}^{*}, \dots .

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10.

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Write out the elements of each of the sets

\begin{aligned} P (\emptyset), & P (P (\emptyset)), & P (P (P (\emptyset))), & P (P (P (P (\emptyset)))) . \end{aligned}

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Make sure you have all the pairs of braces

{}

you should have.

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Without computing it, make a conjecture about the number of elements in the set

P (P (P (P (P (\emptyset))))) .

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Properties of power sets.

For each of Exercises 11–14, either formally prove the given statement about power sets or demonstrate that it is false by providing a specific counterexample.

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